Your question also applies to alpha particles.
Some science authors may use doubly-ionized helium nuclei (He2+) and alpha particles as interchangeable terms. Thus, alpha particles may be loosely used as a term when referring to stellar helium nuclei reactions (for example the alpha processes), and even when they occur as components of cosmic rays. However, helium nuclei produced by particle accelerators (cyclotrons, synchrotrons, and the like) are less likely to be referred to as "alpha particles" because the high energies produced by these sources highlights the striking difference in behavior of their particles from the classical alpha particles produced (and originally defined by) the process of radioactive alpha decay.
The same goes for a proton beam.
I think a good guess, is that if you're dealing with nuclear physics, call it a proton, anything else call it whatever you want.
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons…
The H+ ion (i.e. proton) is an instance of atom under this definition, as are all ions.
Be sure to check out the parent article, which was inspired by this HN post: